Riemann curvature of a boosted spacetime geometry

TL;DR

This paper numerically evaluates the Riemann curvature tensor of boosted Schwarzschild-de Sitter spacetime, revealing a scalar singularity within a 3-sphere and introducing the concept of a 'boosted horizon' that exhibits antigravity-like effects.

Contribution

First numerical evaluation of the curvature singularity in boosted Schwarzschild-de Sitter spacetime approaching the ultrarelativistic limit.

Findings

Identification of a scalar curvature singularity within a 3-sphere

Discovery of a 'boosted horizon' acting as an elastic, antigravity-like barrier

Numerical demonstration of geodesic behavior avoiding the boosted horizon

Abstract

The ultrarelativistic boosting procedure had been applied in the literature to map the metric of Schwarzschild-de Sitter spacetime into a metric describing de Sitter spacetime plus a shock-wave singularity located on a null hypersurface. This paper evaluates the Riemann curvature tensor of the boosted Schwarzschild-de Sitter metric by means of numerical calculations, which make it possible to reach the ultrarelativistic regime gradually by letting the boost velocity approach the speed of light. Thus, for the first time in the literature, the singular limit of curvature through Dirac's delta distribution and its derivatives is numerically evaluated for this class of spacetimes. Eventually, the analysis of the Kretschmann invariant and the geodesic equation show that the spacetime possesses a scalar curvature singularity within a 3-sphere and it is possible to define what we here call boosted horizon, a sort of elastic wall where all particles are surprisingly pushed away, as numerical analysis demonstrates. This seems to suggest that boosted geometries are ruled by a sort of antigravity effect since all geodesics seem to refuse to enter the boosted horizon, even though their initial conditions are aimed at driving the particles towards the boosted horizon.